Ralph Vince - Portfolio Mathematics

242 THE HANDBOOK OF PORTFOLIO MATHEMATICS

For example, suppose we seek to maximize the product of two numbers,
given that their sum is 20. We will let the variables X and Y be the two
numbers. Therefore, H(X,Y)=X∗Y is the function to be maximized given
the constraining function G(X,Y)=X+Y− 20 =0. We must form the
Lagrangian function:

F(X,Y,L)=X∗Y+L∗(X+Y−20)

FX(X,Y,L)=Y+L

FY(X,Y,L)=X+L

FL(X,Y,L)=X+Y− 20

Now we set FX(X,Y,L) and FY(X,Y,L) both equal to zero and solve each for L:

Y+L= 0

Y=−L

and

X+L= 0

X=−L

Now setting FL(X,Y,L)=0 we obtain X+Y− 20 = 0 .Lastly, we replace X
and Y by their equivalent expressions in terms of L:

(−L)+(−L)− 20 = 0

2 ∗−L= 20

L=− 10

Since Y equals−L, we can state that Y equals 10, and likewise with X. The
maximum product is 10∗ 10 = 100.
The method of Lagrangian multipliers has been demonstrated here for
two variables and one constraint function. The method can also be applied
when there are more than two variables and more than one constraint func-
tion. For instance, the following is the form for finding the extreme when
there are three variables and two constraint functions:

F(X,Y,Z,L 1 ,L 2 )=H(X,Y,Z)+L 1 ∗G 1 (X,Y,Z)+L 2 ∗G 2 (X,Y,Z) (7.08)

In this case, you would have to find the simultaneous solution for five equa-
tions in five unknowns in order to solve for the points of relative extreme.
We will cover how to do that a little later on.
We can restate the problem here as one where we must minimize V, the
variance of the entire portfolio, subject to the two constraints that:
(N
∑

i= 1

Xi∗Ui

)

−E= 0 (7.09)